Sets

Sets

Question

Prove the following by using the principle of mathematical induction for all space space straight n element of straight N.

1.3 space plus space 2.3 squared space plus space 3.3 cubed space plus space........ space plus space straight n.3 to the power of straight n space equals space fraction numerator left parenthesis 2 straight n minus 1 right parenthesis space 3 to the power of straight n plus 1 end exponent space plus space 3 over denominator 4 end fraction


Answer

Let P(n): 1.3 space plus space 2.3 squared space plus space 3.3 cubed space plus space.......... space plus space straight n.3 to the power of straight n space equals space fraction numerator left parenthesis 2 straight n minus 1 right parenthesis to the power of 3 straight n plus 1 end exponent plus 3 over denominator 4 end fraction
I.         For n = 1,
         P(1) : 1.3 space equals space fraction numerator left square bracket 2 left parenthesis 1 right parenthesis minus 1 right square bracket 3 to the power of 1 plus 1 end exponent plus 3 over denominator 4 end fraction space rightwards double arrow space 3 space equals space fraction numerator 1 left parenthesis 3 squared right parenthesis plus 3 over denominator 4 end fraction space rightwards double arrow space 3 space equals space fraction numerator 9 plus 3 over denominator 4 end fraction space rightwards double arrow space 3 space equals space 3

∴       P(1) is true.
II.      Suppose the statement is true for n = m, straight m element of space straight N
         P(m): space space 1.3 space plus space 2.3 squared plus 3.3 cubed plus..... space plus space straight m.3 to the power of straight m space equals space fraction numerator left parenthesis 2 straight m minus 1 right parenthesis 3 to the power of straight m plus 1 end exponent plus 3 over denominator 4 end fraction     ...(i)
III.     For n = (m + 1),
         P(m + 1) : 1.3 space plus space 2.3 squared plus 3.3 cubed space plus space......... space plus space left parenthesis straight m plus 1 right parenthesis space 3 to the power of straight m plus 1 end exponent space equals space fraction numerator open square brackets 2 left parenthesis straight m plus 1 right parenthesis space minus 1 close square brackets 3 to the power of straight m plus 1 plus 1 end exponent plus 3 over denominator 4 end fraction
or        1.3 space plus space 2.3 squared space plus space 3.3 cubed space plus space....... plus space straight m.3 to the power of straight m plus left parenthesis straight m plus 1 right parenthesis 3 to the power of straight m plus 1 end exponent space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis.3 to the power of straight m plus 2 end exponent plus 3 over denominator 4 end fraction
      From (i),
         <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>

∴   <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
rightwards double arrow space space space fraction numerator left parenthesis 2 straight m minus 1 right parenthesis space 3 to the power of straight m plus 1 end exponent over denominator 4 end fraction plus space 3 over 4 space plus space left parenthesis straight m plus 1 right parenthesis space 3 to the power of straight m plus 1 end exponent space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis 3 to the power of straight m plus 2 end exponent space plus space 3 over denominator 4 end fraction
rightwards double arrow space space space space space space 3 to the power of straight m plus 1 end exponent open square brackets fraction numerator 2 straight m minus 1 over denominator 4 end fraction plus left parenthesis straight m plus 1 right parenthesis close square brackets space plus space 3 over 4 space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis 3 to the power of straight m plus 2 end exponent plus 3 over denominator 4 end fraction
space space rightwards double arrow space space space space 3 to the power of straight m plus 1 end exponent open square brackets fraction numerator 6 straight m plus 3 over denominator 4 end fraction close square brackets space plus space 3 over 4 space equals space fraction numerator left parenthesis 2 straight m space plus space 1 right parenthesis 3 to the power of straight m plus 2 end exponent space plus space 3 over denominator 4 end fraction space rightwards double arrow space fraction numerator 3 to the power of straight m plus 1 end exponent.3 left parenthesis 2 straight m space plus 1 right parenthesis over denominator 4 end fraction space plus 3 over 4 space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis.3 to the power of straight m plus 2 end exponent space plus 3 over denominator 4 end fraction
rightwards double arrow space space space space space space space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis to the power of 3 straight m plus 2 end exponent plus 3 over denominator 4 end fraction space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis 3 to the power of straight m plus 2 end exponent plus 3 over denominator 4 end fraction
         Which is true

∴        P(m + 1) is true

∴        P(m) is true rightwards double arrow P(m + 1) is true
Hence, by the principle of mathematical induction, statement P(n) is true for all straight n space element of space straight N.


More Chapters from Sets